StratonovichProcess

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`StratonovichProcess`

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StratonovichProcess[{a,b},x,t]

represents a Stratonovich process , where .

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StratonovichProcess[{a,b,c},x,t]

represents a Stratonovich process , where .

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StratonovichProcess[…,…,{x,x₀},{t,t₀}]

represents a Stratonovich process with initial condition .

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StratonovichProcess[…,…,…,Σ]

uses a Wiener process , with covariance Σ.

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StratonovichProcess[proc]

converts proc to a standard Stratonovich process whenever possible.

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StratonovichProcess[sdeqns,expr,x,t,wdproc]

represents a Stratonovich process specified by a stochastic differential equation sdeqns, output expression expr, with state x and time t, driven by w following the process dproc.

Details and Options

StratonovichProcess is also known as Stratonovich diffusion or stochastic differential equation (SDE).
StratonovichProcess is a continuous-time and continuous-state random process.
If the drift a is an -dimensional vector and the diffusion b an ×-dimensional matrix, the process is -dimensional and driven by an -dimensional WienerProcess.
Common specifications for coefficients a and b include:
a scalar, b scalar

a scalar, b vector

a vector, b vector

a vector, b matrix
A stochastic differential equation is sometimes written as an integral equation .
The default initial time t₀ is taken to be zero, and default initial state x₀ is zero.
The default covariance Σ is the identity matrix.
A standard Stratonovich process has output , consisting of a subset of differential states .
Processes proc that can be converted to standard StratonovichProcess form include OrnsteinUhlenbeckProcess, GeometricBrownianMotionProcess, ItoProcess, and StratonovichProcess.
The stochastic differential equations in sdeqns can be of the form , where is \[DifferentialD], which can be input using dd. The differentials and are taken to be Stratonovich differentials.
The output expression expr can be any expression involving x[t] and t.
The driving process dproc can be any process that can be converted to a standard Stratonovich process.
Properties related to StratonovichProcess include:

	"Drift"	drift term
	"Diffusion"	diffusion matrix
	"Output"	output state
	"TimeVariable"	time variable
	"TimeOrigin"	origin of time variable
	"StateVariables"	state variables
	"InitialState"	initial state values
	"KolmogorovForwardEquation"	Kolmogorov forward equation (Fokker-Planck equation)
	"KolmogorovBackwardEquation"	Kolmogorov backward equation
	"Derivative"	Stratonovich derivative

Method settings in RandomFunction specific to StratonovichProcess include: »

	"EulerMaruyama"	Euler–Maruyama (order 1/2, default)
	"KloedenPlatenSchurz"	Kloeden–Platen–Schurz (order 3/2)
	"Milstein"	Milstein (order 1)
	"StochasticRungeKutta"	3‐stage Rossler SRK scheme (order 1)
	"StochasticRungeKuttaScalarNoise"	3‐stage Rossler SRK scheme for scalar noise (order 3/2)

StratonovichProcess can be used with such functions as RandomFunction, CovarianceFunction, PDF, and Expectation.

Examples

open allclose all

Basic Examples (1)Summary of the most common use cases

Define a process by its stochastic differential equation:

In[1]:=1

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https://wolfram.com/xid/0gcpasm4u52a8m450i-erc776

Out[1]=1

Simulate the process:

In[3]:=3

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https://wolfram.com/xid/0gcpasm4u52a8m450i-cdn36k

Out[3]=3

In[4]:=4

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https://wolfram.com/xid/0gcpasm4u52a8m450i-fa7em1

Out[4]=4

Compute the mean function:

In[5]:=5

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https://wolfram.com/xid/0gcpasm4u52a8m450i-fypbq9

Out[5]=5

Compute the covariance function:

In[6]:=6

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https://wolfram.com/xid/0gcpasm4u52a8m450i-bmlcrb

Out[6]=6

In[7]:=7

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https://wolfram.com/xid/0gcpasm4u52a8m450i-f4s7c4

Out[7]=7

Scope (16)Survey of the scope of standard use cases

Basic Uses (10)

Define a Wiener process with drift and diffusion from the sde :

In[1]:=1

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https://wolfram.com/xid/0gcpasm4u52a8m450i-7oicj

Out[1]=1

Directly convert from the parametric process:

In[2]:=2

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https://wolfram.com/xid/0gcpasm4u52a8m450i-gnejul

Out[2]=2

Define a process , where :

In[1]:=1

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https://wolfram.com/xid/0gcpasm4u52a8m450i-ide2rk

Out[1]=1

Use differential notation to define the same process:

In[2]:=2

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https://wolfram.com/xid/0gcpasm4u52a8m450i-3pa1k

Out[2]=2

Define a vector process with output :

In[1]:=1

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https://wolfram.com/xid/0gcpasm4u52a8m450i-b1gly5

Out[1]=1

Using differential notation:

In[2]:=2

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https://wolfram.com/xid/0gcpasm4u52a8m450i-giqh4

Out[2]=2

Define a vector process , where :

In[1]:=1

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https://wolfram.com/xid/0gcpasm4u52a8m450i-clajfc

Out[1]=1

Using differential notation:

In[2]:=2

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https://wolfram.com/xid/0gcpasm4u52a8m450i-bm8t6

Out[2]=2

Define a vector process where :

In[1]:=1

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https://wolfram.com/xid/0gcpasm4u52a8m450i-jqo1f

Out[1]=1

Using differential notation:

In[2]:=2

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https://wolfram.com/xid/0gcpasm4u52a8m450i-lhcb27

Out[2]=2

Define a process driven by two correlated Wiener processes:

In[1]:=1

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https://wolfram.com/xid/0gcpasm4u52a8m450i-dp6cu3

In[2]:=2

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https://wolfram.com/xid/0gcpasm4u52a8m450i-hdxbmx

Out[2]=2

Define a scalar process corresponding to the sde :

In[1]:=1

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https://wolfram.com/xid/0gcpasm4u52a8m450i-jjkzid

Out[1]=1

Define vector process and corresponding to the sde and :

In[1]:=1

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https://wolfram.com/xid/0gcpasm4u52a8m450i-dpsd7d

Out[1]=1

Define a process corresponding to the 2D correlated Wiener process:

In[1]:=1

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https://wolfram.com/xid/0gcpasm4u52a8m450i-b3q1e

Define a vector process driven by the correlated 2D Wiener process:

In[2]:=2

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https://wolfram.com/xid/0gcpasm4u52a8m450i-ivsi62

Out[2]=2

Simulate StratonovichProcess paths using different methods:

In[1]:=1

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https://wolfram.com/xid/0gcpasm4u52a8m450i-qudz7b

Out[1]=1

Simulation methods and their corresponding orders:

In[2]:=2

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https://wolfram.com/xid/0gcpasm4u52a8m450i-hdg78k

In[3]:=3

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https://wolfram.com/xid/0gcpasm4u52a8m450i-e7n7ir

Specify the simulation method as an option in RandomFunction:

In[4]:=4

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https://wolfram.com/xid/0gcpasm4u52a8m450i-zh8ux

In[5]:=5

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https://wolfram.com/xid/0gcpasm4u52a8m450i-hpm9od

Out[5]=5

Process Properties Extraction (1)

Define a Stratonovich process by its stochastic differential equation:

In[1]:=1

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https://wolfram.com/xid/0gcpasm4u52a8m450i-fi3b0g

Out[1]=1

Available Stratonovich process properties:

In[2]:=2

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https://wolfram.com/xid/0gcpasm4u52a8m450i-oi3itr

Out[2]=2

Drift and diffusion of the process:

In[3]:=3

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https://wolfram.com/xid/0gcpasm4u52a8m450i-lhoaz

Out[3]=3

Kolmogorov forward equation:

In[4]:=4

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https://wolfram.com/xid/0gcpasm4u52a8m450i-jzoz3

Inactive is used here to avoid expanding the partial derivatives; use Activate to expand the expression:

In[5]:=5

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https://wolfram.com/xid/0gcpasm4u52a8m450i-b31i1a

Kolmogorov backward equation:

In[6]:=6

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https://wolfram.com/xid/0gcpasm4u52a8m450i-cpz1oy

Compute the Stratonovich derivative of a function . The output is a list consisting of drift and diffusion terms:

In[7]:=7

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https://wolfram.com/xid/0gcpasm4u52a8m450i-ir1l41

In[8]:=8

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https://wolfram.com/xid/0gcpasm4u52a8m450i-oo5n0g

In[9]:=9

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https://wolfram.com/xid/0gcpasm4u52a8m450i-e2szkg

Special Stratonovich Processes (5)

A Stratonovich process corresponding to the WienerProcess:

In[1]:=1

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https://wolfram.com/xid/0gcpasm4u52a8m450i-lcml5n

Out[1]=1

A Stratonovich process corresponding to the GeometricBrownianMotionProcess:

In[1]:=1

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https://wolfram.com/xid/0gcpasm4u52a8m450i-f88nca

Out[1]=1

A Stratonovich process corresponding to the BrownianBridgeProcess:

In[1]:=1

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https://wolfram.com/xid/0gcpasm4u52a8m450i-ewpvdh

Out[1]=1

A Stratonovich process corresponding to the OrnsteinUhlenbeckProcess:

In[1]:=1

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https://wolfram.com/xid/0gcpasm4u52a8m450i-id518c

Out[1]=1

A Stratonovich process corresponding to the CoxIngersollRossProcess:

In[1]:=1

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https://wolfram.com/xid/0gcpasm4u52a8m450i-b38md3

Out[1]=1

Applications (3)Sample problems that can be solved with this function

Define a vector process corresponding to iterated Stratonovich integrals , , , , , :

In[1]:=1

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https://wolfram.com/xid/0gcpasm4u52a8m450i-cu3zkl

Out[1]=1

Compute its mean function:

In[2]:=2

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https://wolfram.com/xid/0gcpasm4u52a8m450i-eivp8l

Out[2]=2

And its covariance function:

In[3]:=3

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https://wolfram.com/xid/0gcpasm4u52a8m450i-f7odx9

The dynamics of a free particle under the effect of thermal fluctuation can be modeled by the Langevin equation of motion, , where is the standard WienerProcess and is the strength of the thermal noise. Here it is assumed that can only depend on and focus on the equation of velocity. There are two common ways to integrate the equation of motion: Ito formulation and Stratonovich formulation:

In[1]:=1

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https://wolfram.com/xid/0gcpasm4u52a8m450i-bern42

In[2]:=2

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https://wolfram.com/xid/0gcpasm4u52a8m450i-ezo8v4

When is a constant, the two formulations are identical and lead to the same stationary distribution as :

In[3]:=3

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https://wolfram.com/xid/0gcpasm4u52a8m450i-cm474u

Out[3]=3

In[4]:=4

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https://wolfram.com/xid/0gcpasm4u52a8m450i-wqaem

Out[4]=4

If is velocity dependent, then due to the nature of the WienerProcess, has nonzero quadratic variation and the two formulations lead to different results. Convert Ito formulation to the equivalent Stratonovich formulation: